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SOME DEFINABLE GALOIS THEORY AND EXAMPLES
OMAR LEÓN SÁNCHEZ AND ANAND PILLAY
Abstract. We make explicit certain results around the Galois correspondence
in the context of definable automorphism groups, and point out the relation
to some recent papers dealing with the Galois theory of algebraic differential
equations when the constants are not “closed” in suitable senses. We also
improve the definitions and results from [15] on generalized strongly normal
extensions, using this to give a restatement of a conjecture from [1].
1. Introduction
We begin this introduction with some comments for the general logician, explaining what the paper is about and putting it in context.
The key notions are Galois theory and the Galois correspondence. The basic
example is taking a field K (such as the rationals) and adjoining to it all roots of
some polynomial over K in one variable, to generate a field L say. L is then said to
be a Galois extension of K. The (finite) group G of automorphisms of L which fix
K pointwise is called the Galois group of L over K and the Galois correspondence is
between subgroups of G and fields in between K and L. The paper [13] puts this into
a more general model theoretic framework, following Poizat [19]. Another example,
closer to the topic of this paper is when K is a differential field (i.e., a field equipped
with a derivation ∂, such as the field of rational functions f (t) over a field k together
with the derivation d/dt) and in place of a polynomial equation over K we are given
a linear differential equation over K: ∂ (n) y + an−1 ∂ (n−1) y + ... + a1 ∂y + a0 y = 0.
Rather than adjoining all solutions to K we want to adjoin a so-called fundamental
system of solutions, to form a differential field L such that the field of constants of
L is the same as the field of constants of K. L is called a Picard-Vessiot extension
of K. When the field of constants CK of K is algebraically closed, this can be
accomplished. Moreover, in this case the group G of automorphisms of L over K
(as differential field), which may no longer be finite, is naturally isomorphic to a
subgroup of GLn (CK ) defined by polynomial equations (a so-called linear algebraic
group), where GLn (CK ) denotes the group of n×n matrices of nonzero determinant,
with entries from CK . The Galois correspondence is between algebraic subgroups of
G and differential fields in between K and L. The same paper of Poizat referred to
above gives a model theoretic account of this Picard-Vessiot Galois theory, making
use of the first order theory DCF0 of differentially closed fields of characteristic
zero, among other things. There are many variants of the Picard-Vessiot theory,
which are detailed below. For example, we can consider a field K equipped with
Date: August 18, 2016.
2010 Mathematics Subject Classification. 03C60, 12H05.
Key words and phrases. model theory, differential fields, strongly normal extensions.
Anand Pillay was supported by NSF grant DMS-1360702.
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OMAR LEÓN SÁNCHEZ AND ANAND PILLAY
two commuting derivations, sometimes called ∂x and ∂t , and a linear differential
equation over K involving only the derivation ∂x . When the field CK of ∂x -constants
of K is differentially closed as a ∂t -field, one again obtains the analogous existence of
a Picard-Vessiot extension L of K, a Galois group G, and a Galois correspondence,
except that now G is a subgroup of GLn (CK ) defined by differential equations
involving ∂t .
Several papers and/or preprints have appeared recently, giving a version of the
Galois correspondence when the “constants” of the base field are not necessarily
“closed” in the appropriate senses. The point of our current paper is to explain
how these results are implicit or explicit in the model-theoretic literature: see [6,
Section B.4] by Hrushovski or [7, Section 2.2] by Kamensky. This is parts (iii), (iv),
and (v) of Theorem 2.2 below, a certain “indirect” Galois correspondence in the
context of definable automorphism groups in first order theories.
In fact in the special case of “strongly normal” extensions of a differential field
K whose field of constants CK is not necessarily algebraically closed, the results are
already in Kolchin’s book [8, Chapter VI, Theorem 3]. In all these cases some translation between different languages is required, and the model-theoretic references
may be unknown or rather obscure, at least for differential algebraists.
At this point we assume a little more familiairity with model theory and the
relevant pieces of algebra.
The reason we talk about indirect Galois correspondences is as follows: In usual
Galois theory, we take for K say a perfect field, P (x) = 0 a polynomial equation
over K, and L a splitting field for this equation. We can consider the (finite)
set Y of solutions of the equation in the algebraic closure K alg of K. Then L is
generated over K by Y , and moreover the (finite) group of permutations of Y which
preserve all polynomial relations over K coincides with Aut(L/K). The Galois
correspondence is directly between subgroups of Aut(L/K) and fields in between
K and L. This picture also holds for the Picard-Vessiot theory when the constants
are algebraically closed. Namely we take a linear differential equation δy = Ay over
a differential field (K, δ) of characteristic zero with algebraically closed field CK of
constants. We now let Y be the solution set of the equation in the differential closure
K dif f of K, which is an n-dimensional vector space over the field C of constants
of K dif f . We choose a basis ȳ of Y over C, and let L be the (differential) field
generated over K by ȳ, the so-called Picard-Vessiot extension of K for the equation
δy = Ay. It follows from CK being algebraically closed that actually C = CK .
Hence the full solution set Y is contained in L, and the group of permutations
of Y preserving all differential polynomial relations over K and C, which has the
the structure of an algebraic subgroup G of GLn (C), coincides with Autδ (L/K)
The Galois correspondence is between algebraic subgroups of G = Autδ (L/K)
and differential fields in between K and L, as in the polynomial case. But let
us consider now the differential situation when the field of constants CK of K is
not necessarily algebraically closed. Then C, the algebraic closure of CK , may
properly contain CK . The set Y of solutions of the equation in K dif f is still an
n-dimensional vector space over C, and the group G of permutations of Y which
preserve differential polynomial relations over K and C is an algebraic subgroup of
GLn (C) as before. It may happen that there is a basis ȳ of Y over C such that
the differential field L = F (ȳ) has the same constants as K, in which case L is
called a Picard-Vessiot extension of K for the equation. The existence of such L
SOME DEFINABLE GALOIS THEORY AND EXAMPLES
3
implies that G is defined over CK , and we have moreover that G(C) coincides with
Autδ (L(C)/K(C)) while G(CK ) coincides with Autδ (L/K). There is in general no
direct Galois correspondence between algebraic subgroups of G(CK ) and differential
fields in between K and L. But there is rather a Galois correspondence between
algebraic subgroups of G(C), defined over CK , and differential fields in between
K and L: given such an intermediate differential field F , the elements of G which
fix F pointwise form an algebraic subgroup of G defined over CK , etc. This is an
example of the indirect Galois correspondence, which is given by the existence of a
Picard-Vessiot extension, and which is a special case of the model theoretic result
we will be presenting.
The general model-theoretic context is roughly (working inside an ambient saturated structure M̄ ): we are given a set A of parameters, and A-definable sets Y
and X. Under suitable conditions Aut(Y /A, X), the group of permutations of Y
which are elementary over A ∪ X will have the structure of an A-definable group
G. And under additional conditions we will study the “extension” B/A where
B = dcl(A, b) for a suitable b ∈ Y , as well as Aut(B/A), which are the general versions of a Picard-Vessiot extension L/K, and its automorphism group Aut(L/K).
And we will describe the relations between these objects as well as the (indirect)
Galois correspondence. This will be accomplished in the next section, where the
basic set-up (or definition) is given by conditions (I) and (II) and the main result
is Theorem 2.3. As remarked in [6] what is being discussed is when and how the
theory of definable automorphism groups translates to a theory of extensions B/A
of definably closed sets and their automorphism groups.
In the final section we show how this set-up and Theorem 2.3 subsume the various
differential algebraic examples. One possibly novel thing on the differential algebraic side will be a refinement of the generalized strongly normal theory from [15],
[16], where we are able to drop the assumption that X(K) = X(K dif f ). We will use
this to give a rather clearer statement of Conjecture 2.3 from [1], which is related to
Ax-Lindemann-Weierstrass questions for (families of) semiabelian varieties. This
is possibly the only part of the paper which requires some more background from
the reader.
We should make it clear that our aim is not to give yet another exposition of
definable automorphism groups and/or the model theoretic approach to the Galois
theory of differential equations, of which there are many ([19], [20, Section 8.3],
as well as the more specialized [18] and [17, Chapter 8, section 4]), but rather to
explain concisely how a number of recent results in the literature follow from general
and known model-theoretic considerations.
2. Model theoretic context and results
Throughout this section we assume familiarity with the basics of model theory
(the material in [12, Chapter 1] covers most of what we use here). Let us fix a
complete theory T in the language L. M̄ denotes a saturated model of T , and A,
B, ... small subsets of M̄ . For convenience we assume
(†)
T is ω-stable and T = T eq
In place of the T = T eq assumption, the reader could just assume that T is
1-sorted with elimination of imaginaries, which will actually be the case in the
differential algebraic contexts and examples, where T = DCF0,m .
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OMAR LEÓN SÁNCHEZ AND ANAND PILLAY
We fix a definably closed set A, a complete type q(x) over A, and an A-definable
set X. We consider the following conditions on q and X.
(I) For any realizations b, b0 of q, b0 ∈ dcl(b, X, A).
(II) For any tuple c from X, q(x) and r(y) = tp(c/A) are weakly orthogonal,
namely q(x) ∪ r(y) extends to a unique complete xy-type over A.
We do not really wish to add new and unnecessary terminology to the subject,
but we mention some possibilities in the following remark.
Remark 2.1.
(1) Condition (I) says that q is internal to X and is already the type of a “fundamental system of solutions”. We could use the expression “q is strongly
internal to X” for (I).
(2) Condition (II) says that for any two realizations b1 , b2 of q in M̄ ,
tp(b1 /A, X) = tp(b2 /A, X),
which has to be the unique nonforking extension of q over A ∪ X.
(3) We might also want to express conditions (I) and (II) by “B is a Galois
extension of A, relative to X” where B = dcl(A, b) and b realizes q.
Lemma 2.2.
(i) Let b realize q and let B = dcl(A, b). Then q and X satisfy condition (II)
above if and only if
(II’) dcl((A ∪ X) ∩ B) = A.
(ii) Suppose (I) and (II) hold (for q, X). Then q is isolated.
Proof. (i) (II) implies (II’) is clear. For the converse: By ω-stability there is c ∈
dcl(A, X) such that tp(b/A, X) is definable over c and note that c ∈ dcl(b, A). So
c ∈ dcl((A ∪ X) ∩ B). By (II’), c ∈ A, which means that tp(b/A, X) is definable
over A. Note that for any other realization b0 of q there is an A-automorphism of
M̄ taking b to b0 and fixing X setwise, hence tp(b0 /A ∪ X) is definable over A by the
same schema defining tp(b/A ∪ X), and hence tp(b0 /A ∪ X) = tp(b/A ∪ X), giving
(II).
(ii) By (I) and compactness there is a formula φ(x) ∈ q(x) and a partial A-definable
function f (−, −) such that for all b, b0 satisfying φ(x), there is a tuple c of elements
of X such that f (b, c) = b0 . Now, by ω-stability, let M0 be a prime model over
A ∪ X, and let b realize φ(x) in M0 and let b0 be any realization of q in M̄ . By what
we just said there is c from X such that f (b, c) = b0 . As tp(b/A, X) is isolated, so is
tp(b0 /A, X). But by (II), q has a unique extension to a complete type over A ∪ X,
which must be tp(b0 /A, X). By compactness q is isolated.
We are still fixing A, q and X. Let Q be the set of realizations of q (in M̄ ). Let
Aut(Q/X, A) denote the group of permutations of Q which are elementary over
A ∪ X, namely preserve all relations (on arbitrary X n ) which are A ∪ X-definable.
Equivalently (using stability), Aut(Q/X, A) is the group of permutations of X
induced by automorphisms of M̄ which fix pointwise A ∪ X. Let us fix a realization
b of q, and let B = dcl(A, b). Let Aut(B/A) denote the group of permutations of
B which are elementary over A in the sense of M̄ . With this notation we have:
Theorem 2.3. Suppose that q and X satisfy conditions (I) and (II). Then
SOME DEFINABLE GALOIS THEORY AND EXAMPLES
5
(i) Aut(Q/A, X) acts regularly (in particular transitively) on Q. Indeed every
σ ∈ Aut(Q/A, X) is determined by σ(b0 ) for some/any b0 ∈ Q and all
elements of Q have the same type over A ∪ X.
(ii) There is an A-definable group G whose domain is an A-definable set contained in dcl(A, X), and a group isomorphism µ between Aut(Q/A, X) and
G such that the induced action of G on Q is B-definable.
(iii) µ induces an isomorphism between Aut(B/A) and G(A), the group of elements of G which are in A.
(iv) There is a Galois correspondence between definably closed sets in between
A and B and A-definable subgroups of G. More precisely, for a definably
closed set C with A ⊆ C ⊆ B, if we let
HC = {σ ∈ G : σ fixes C pointwise},
then HC is an A-definable subgroup of G, every A-definable subgroup of G
appears as some HC , and C ⊆ C 0 iff HC 0 ≤ HC . In addition:
(v) Let C be a definably closed set with A ⊆ C ⊆ B. Then, then tp(b/C) and X
also satisfy conditions (I) and (II), and HC is isomorphic to Aut(QC /C, X),
where QC is the set of realizations of tp(b/C). Moreover, there is a finite
tuple e such that C = dcl(e, A), and the type tp(e/A) satisfies (I) if and
only if HC is a normal subgroup of G. In this case, G/HC is isomorphic
to Aut(P/A, X) where P is the set of realizations of tp(e/A).
Proof. Everything is explicit or implicit in the literature, but we give a quick proof
of the theorem in its entirety. (i) and (ii) set up the objects, and the main points
are (iii), (iv), and (v).
The proof of (i) is contained in the statement.
(ii) In Lemma 2.2 we saw that
(*)
Q is an A-definable set,
and moreover by (II) (and the proof of Lemma 2.1 (i))
(**)
q has a unique extension to a complete type over A ∪ X
which is moreover definable over A.
By (I) and compactness, there is an A-definable (partial) function f0 such that
every element of Q is of the form f0 (b, c) for some tuple c (of fixed length) from X.
Let Y0 be the set of tuples of this fixed length from X such that f0 (b, c) ∈ Q. By
(*) and (**) Y0 is an A-definable set of tuples from X. Let E be the equivalence
relation on Y0 defined by E(c1 , c2 ) iff f0 (b, c1 ) = f0 (b, c2 ). Then, again by (**),
E is A-definable. Hence the set Y0 /E which is A-definable in M̄ eq is contained in
dcl(A, X). Let Y denote Y0 /E. We can clearly rewrite f0 as an A-definable map
from Q × Y to Q and now note that for any b0 ∈ Q there is a unique d ∈ Y such
that b0 = f (b, d).
For σ ∈ Aut(Q/A, X) let µ(σ) be the unique element of Y such that σ(b) =
f (b, µ(σ)). The map µ sets up a bijection between Aut(Q/A, X) and Y . Using
(**) one sees that the group operation on Y induced by µ is A-definable, and the
induced action on Q is B-definable. We let G denote the set Y equipped with this
group structure. So G is an A-definable group and its action on Q is definable over
B. We will often identify G with Aut(Q/A, X).
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OMAR LEÓN SÁNCHEZ AND ANAND PILLAY
(iii) As mentioned above G lives on the A-definable set Y , so G(A) is as a set the
collection of elements of Y which are in dcl(A) = A. The nature of our identification
of G with Aut(Q/X, A) yields that G(A) consists of those σ ∈ Aut(Q/X, A) such
that σ(b) ∈ B. Here is the explanation: σ ∈ G(A) means µ(σ) ∈ A which implies
σ(b) = f (b, µ(σ)) ∈ B. Conversely, if σ(b) ∈ B then as µ(σ) is determined by b
and σ(b), also µ(σ) ∈ B. But µ(σ) ∈ Y ⊂ dcl(A, X), hence by Lemma 2.2 and
condition (II), µ(σ) ∈ A, as required. Note finally that we can view Aut(B/A) as
precisely the subgroup of Aut(Q/X, A) consisting of those σ such that σ(b) ∈ B.
This completes the proof.
One should note that as by (II)’, dcl(A, X) ∩ B = dcl(A, X) ∩ A, G(B) coincides
with G(A).
(iv) For the Galois correspondence the main point is that condition (II) yields that
(***)
for D1 , D2 ⊆ B,
D2 ⊆ dcl(D1 , A) iff D2 ⊆ dcl(D1 , A, X).
Note that the latter is equivalent to “D2 is fixed by every σ ∈ G which fixes D1 ”.
Moreover, for d ∈ B
(****) {σ ∈ G : σ(d) = d} is an A-definable subgroup of G.
This is because if d = h(b) for h an A-definable function, then σ ∈ G fixes d iff
h(σ(b)) = h(b) iff h(f (b, σ)) = h(b), which is, by (**), an A-definable condition on
σ. Now let C be an arbitrary subset of B and HC the set of σ ∈ G which fix C
pointwise. By (****) and ω-stability, HC is an A-definable subgroup of G and, by
(***), the set of elements of B which are fixed by HC is precisely dcl(A, C).
On the other hand, suppose H is an arbitrary subgroup of G and let C be the set
of elements of B fixed pointwise by every element of H. Then, as the elements of H
also fix A-pointwise, it follows that C is a definably closed subset of B containing
A. Moreover, by the above, HC is an A-definable subgroup of G containing H.
(v) Let qC = tp(b/C), and let QC denote its set of realizations. As qC extends q,
(I) holds for qC and X. As B = dcl(A, b), tp(B/A) has a unique extension over
A ∪ X, so tp(bC/A) has a unique extension over A ∪ X, whereby tp(b/C) has a
unique extension over C ∪ X. So qC and X satisfy (II).
As remarked earlier any σ ∈ Aut(Q/A ∪ X) is determined by the value σ(b).
Hence {σ ∈ Aut(Q/A ∪ X) : tp(σ(b)/C) = tp(b/C)} is a subgroup of Aut(Q/A ∪ X)
which is naturally isomorphic to Aut(QC /A ∪ X). Moreover the image of this
subgroup under µ is clearly HC .
As C is a definably closed subset of dcl(A, b), ω-stability implies that C =
dcl(A, e) for some finite tuple e ∈ C. Note that p = tp(e/A) (and X) automatically satisfy condition (II). Suppose that tp(e/A) and X satisfy (I). Let σ ∈
Aut(QC /A ∪ X) and τ ∈ Aut(Q/A ∪ X). Let e0 = τ (e). So by (I), e0 ∈ dcl(A, e, X),
whereby σ(e0 ) = e0 . Hence σ −1 τ σ(e) = e, so σ −1 τ σ is in Aut(QC /A ∪ X). We have
shown that Aut(QC /A∪X) is normal in Aut(Q/A∪X), whence HC is normal in G.
Conversely, if Aut(QC /A∪X) is normal in Aut(Q/A∪X), then for any ρ ∈ HC and
σ ∈ G we have ρ(σ(e)) = σ(e). Therefore, all the elements of HC fix P pointwise
(the latter is the set of realizations of p). Thus, P ⊆ dcl(e, A, X). This shows that
p and X satisfy (I). Finally, the restriction to P yields a group homomorphism from
G to Aut(P/A, X) with kernel HC .
SOME DEFINABLE GALOIS THEORY AND EXAMPLES
7
We end this section with a couple of remarks. The first concerns the definability
properties of G(A). G is of course a group definable in the ambient structure M̄ ,
with parameters from A. What about G(A) which corresponds to Aut(B/A)? We
maintain the earlier assumptions, notation and conditions (I), (II), for q, X.
Remark 2.4.
(1) Let M0 be the prime model over A (which exists by ω-stability). Suppose
that X(A) = X(M0 ), namely all points of X in the model M0 are already
in A. Then G(A) coincides with the interpretation G(M0 ) of the formula
over A defining G in M̄ , in the elementary substructure M0 . So bearing
in mind part (iii) of Theorem 2.3, we see in this case that Aut(B/A) is a
definable group in the model M0 , and we easily obtain from Theorem 2.3
the Galois correspondence between subgroups of G(M0 ) definable in M0
and definably closed sets in between A and B.
(2) If the theory T is one-sorted (with elimination of imaginaries) and has
quantifier elimination, then X will be a quantifier-free A-definable set in
M̄ , hence as the universe of G is Y which will be an A-definable subset of
some power of X, G(A) will be a group (quantifier-free) definable (without
parameters) in the structure whose universe is X(A) and whose relations are
intersections with X(A)n of A-definable subsets of X n . See Example 3.4 in
Section 3 for the failure of the direct Galois correspondence in this situation.
In the next remark we point out an extension of part (iii) of Theorem 2.3, whose
proof is left to the reader.
Remark 2.5. Under the same assumptions and notation of Theorem 2.3, let D be
any definably closed set containing B, and let BD be dcl(b, A, X(D)). Then µ
induces an isomorphism between Aut(BD /A, X(D)) and G(D).
3. Applications to and interpretation in differential Galois theory
We now give the interpretation of the above model-theoretic results in the context
of differential fields. In these examples we will start with the most concrete, ending
with the most general, for pedagogical reasons.
We work in the language of differential rings with m derivations
Lrings ∪ {δ1 , . . . , δm }.
All model-theoretic and differential-algebraic terminology refers to this language.
Recall that DCF0,m is the model companion of the theory of fields of characteristic
zero equipped with m commuting derivations, and is called the theory of differentially closed fields of characteristic zero (with m commuting derivations). It is
well known that this theory is ω-stable and eliminates imaginaries, so fits into the
assumption (†). Moreover, it has quantifier elimination [14]. When m = 1 (the socalled ordinary case) the theory is sometimes just denoted DCF0 . In this ordinary
case we write δ for δ1 .
The saturated model M̄ is denoted
(U, Π = {δ1 , . . . , δm }) |= DCF0,m .
K, L, etc. denote (small) differential subfields K of U. For any set A of tuples from
U, the definable closure of A in U coincides with the differential field generated by
the coordinates of the members of A, and the (model-theoretic) algebraic closure
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OMAR LEÓN SÁNCHEZ AND ANAND PILLAY
of A coincides with the field-theoretic algebraic closure of the differential field just
mentioned. KhAi denotes the differential field generated over K by A, where A is
a subset of U. For any subset D ⊆ Π and any differential subfield F of U we let
F D denote the subfield of D-constants of F ; that is,
F D = {a ∈ F : δ(a) = 0 for all δ ∈ D}.
We let C denote U Π the universal field of absolute constants, and also write
CF for F Π (the absolute constants of F ). This is consistent with notation in the
introduction.
Lemma 3.1. Suppose K ⊆ L are differential fields (i.e. differential subfields of
U). Let D be a nonemptyset subset of Π, and let X = U D . Then
(1) dcl(K ∪ X) ∩ L = KhLD i.
(2) dcl(K ∪ X) ∩ L = K means precisely that K D = LD .
Proof. (i) A consequence of the linear disjointness property of the constants (see
[8, Chap. II, §1]) is that for any intermediate differential field K ≤ E ≤ KhU D i,
we have that
KhE D i = E.
Applying this with E = dcl(K ∪ X) ∩ L we get
dcl(K ∪ X) ∩ L = KhU D i ∩ L = KhKhU D i ∩ LD i = KhLD i.
(ii) follows directly from (i).
3.1. Picard-Vessiot extensions. We focus first on the ordinary case, m = 1.
A linear differential equation over K (in vector form) is something of the form
δy = Ay where y is a n × 1 column vector of indeterminates and A is an n × n
matrix with coefficients from K. A fundamental system of solutions of the equation
is a set y1 , .., yn of solutions of the equation such that the determinant of the n × n
matrix (y1 , .., yn ) is nonzero. Equivalently (y1 , .., yn ) is a basis over C of the C-vector
space V of solutions to the equation in U. By a Picard-Vessiot extension of K for
the equation δy = Ay is meant a differential field L generated over K by some
fundamental system (y1 , .., yn ) of solutions, and such that CL = CK . So we see
from Lemma 3.1 that if L is a Picard-Vessiot extension of K for the equation then
L is of the form dcl(K, b), where q = tp(b/K) and C satisfy conditions (I) and (II)
from the previous section. Hence Theorem 2.3 holds in this specific context. In this
case the construction or identification of the Galois group is more direct: for each
σ ∈ Aut(Q/A, C), σ(b) = bµ(σ) for a unique matrix µ(σ) ∈ GL(n, C), and µ gives
an isomorphism between Aut(Q/A, C) and an algebraic subgroup G of GL(n, C)
defined over CL = CK . The Galois correspondence is between algebraic subgroups
of G defined over CK and differential fields in between K and L. Theorem 2.3
in this context is contained in [8, Chap. VI] in the slightly more general form of
strongly normal extensions. In any case what we have described here subsumes [4,
Theorem 4.4] and [3].
Everything extends to the partial case m > 1, where we now consider a set
δ1 y = A1 y, ...., δm y = Am y of linear DE’s over K and where the matrices Ai
satisfy suitable integrability (or Frobenius) conditions.
SOME DEFINABLE GALOIS THEORY AND EXAMPLES
9
3.2. Strongly normal extensions. We work with any m ≥ 1. According to
Kolchin [8] a strongly normal extension L of a differential field K is a differential
field extension L of K such that L is finitely generated over K and for every
automorphism σ of U over K, σ(L) ⊆ LhCi and σ fixes CL pointwise. As Kolchin
notes the condition that every automorphism of U over K fixes CL is equivalent to
requiring that CL = CK .
Let L be generated over K by the finite tuple b, and let q = tp(b/K). Thus the
condition for L to be a strongly normal extension of K means precisely that q and
C satisfy conditions (I) and (II) from the previous section (using Lemma 3.1).
Hence Theorem 2.3 applies. Using elimination of imaginaries, stability, and the
fact that C is an algebraically closed field without additional structure, the Galois
group G as described in the proof of Theorem 2.3, is the group of C-points of an
algebraic group defined over CK . Again the Galois correspondence is between algebraic subgroups of G defined over CK and differential fields in between K and
L. Theorem 2.3 is again contained in Kolchin, [8, Chap. VI]. In fact see Theorems 3 and 4 there for the Galois correspondence. This subsumes [2]. Moreover
part (iii) that Aut(L/K) is G(A), includes, by virtue of Remark 2.4(2), the statement in [2, Theorem 3.15] that Aut(L/K) is interpretable in the field CK . In fact
one actually obtains definability (rather than just interpretability). Similarly, Remark 2.5 includes the statement in [2, Corollary 3.21] with definability in place of
interpretability.
3.3. Parameterized Picard-Vessiot (P P V ) extensions. The context is a differential field K equipped with m + 1 commuting derivations which we write as
δx , δt1 , ...δtm , and a linear differential equation δx y = Ay over K. Our universal
domain U is a saturated model of DCF0,m+1 , and the solution set V of the equation
is an n-dimensional vector space over U δx , the field of δx -constants of U. A fundamental system of solutions is again a matrix (y1 , .., yn ) of solutions, with nonzero
determinant, and by a P P V -extension of K for the equation we mean a differential
field L generated over K by such a fundamental system of solutions, such that also
K δx = Lδx . If (y1 , .., yn ) is such and its type over K is q then as before, q and U δx
satisfy conditions (I) and (II). The Galois group G from Theorem 2.3 is a subgroup
of GLn (U δx ) definable over K δx in the differentially closed field (U δx , δt1 , ..., δtm ).
The Galois correspondence (when K δx is not necessarily differentially closed) was
established in Section 8.1 of [5], but again is contained in Theorem 2.3.
A strongly normal analogue of P P V extensions was studied in [11] also establishing the Galois correspondence.
3.4. Landesman’s strongly normal extensions. Landesman [9] slightly generalizes Kolchin’s definition of strongly normal extension by replacing C(= U Π ) by
U D for any nonempty subset D of Π. Formally he takes ∆ to be Π \ D and defines
L to be a ∆-strongly normal extension of K if L is finitely generated over K, and
for any automorphism σ of U over K, σ(L) ⊆ LhU D i and σ fixes LD pointwise.
As in our discussion of strongly normal extensions above, this is equivalent to requiring that q, U D satisfy conditions (I) and (II) from Section 2. In [9, Theorem
2.34] Landesman proves the Galois correspondence when K D is differentially closed
as a ∆-field, and asks if the differentially closed assumption can be dropped. Our
Theorem 2.3 says yes (for the indirect Galois correspondence). Note that P P V
extensions are a special case of Landesman strongly normal extensions.
10
OMAR LEÓN SÁNCHEZ AND ANAND PILLAY
3.5. Generalised strongly normal extensions. This is a direct transfer of the
notions in Section 2 to the theory DCF0,m . In the paper [15] the second author
introduced X-strongly normal extensions of a differential field K, in the ordinary
context, where X is an arbitrary K-definable set, also mentioning that things extend to the case of several derivations. The condition that X(K dif f ) = X(K) was
included in the original definition, partly for consistency with the usual account
of the Picard-Vessiot (or strongly normal) theory where the field of constants of
K is assumed to be algebraically closed, and partly so that Autδ (L/K) is a definable group in the differentially closed field K dif f . But in fact the condition that
X(K dif f ) = X(K), can be dropped from Definition 2.1 of [15], still preserving the
Galois correspondence (but now indirect). We briefly explain:
Working now in DCF0,m , given a differential field K, and K-definable set X, we
make:
Definition 3.2. The differential field L is an X-strongly normal extension of K, if
L is finitely generated over K, for each automorphism σ of U over K, σ(L) ⊆ LhXi,
and dcl(K ∪ X) ∩ L = K.
This is precisely Definition 2.1 of [15], adapted to several derivations, and minus
part (i). On the other hand, this also states that q and X satisfy conditions (I)
and (II) from Section 2., where q is the type over K of some generator of L over K.
With this definition, Theorem 2.3 gives the Galois group, as well as the (indirect)
Galois correspondence. The Galois group G is a differential algebraic group, defined
over K, and the Galois correspondence is between K-definable subgroups of G and
differential fields between K and L.
3.6. Logarithmic differential equations on algebraic δ-groups. This is the
only section of the paper which may require some additional background from the
reader. In the m = 1 case, when K is algebraically closed, the generalized strongly
normal extensions of [15] come from “logarithmic differential equations on algebraic
δ-groups”, as explained in [16]. This was suitably generalized by the first author
to the m > 1 case in [10]. In [1] Daniel Bertrand and the second author study
a special class of such algebraic δ-groups and logarithmic differential equations on
them, so-called “almost semiabelian δ-groups”, as part of a term project around
functional Lindemann-Weierstrass theorems for families of semiabelian varieties.
Conjecture 2.3 from [1] says roughly that the Galois group can be identified from
the original equation via “gauge transformations over K”. The aim here is to give
a simple restatement of this property in terms of X-strongly nornal extensions of
K (from Definition 3.2) , for suitable X.
Here we assume m = 1. We refer the reader to [16] or [1] but we repeat some
definitions. By an algebraic δ-group over K we mean an algebraic group G over K
together with a lifting of the dervation δ on K to a derivation s on the structure
sheaf of G. We can and will identify s with a homomorphic rational section over
K of the shifted tangent bundle Tδ (G) → G. The logarithmic derivative dlog(G,s) :
G → LG (to the Lie algebra of G) is the differential algebraic map taking x ∈ G
to δ(x) · s(x)−1 computed in the algebraic group Tδ (G). A logarithmic differential
equation on (G, s) over K is an equation of the form dlog(G,s) (−) = A, where
A ∈ LG(K). When G = GLn and s is the 0-section of T (G) → G, such an
equation is simply a linear differential equation in matrix form.
SOME DEFINABLE GALOIS THEORY AND EXAMPLES
11
(G, s)∂ denotes {x ∈ G : s(x) = δ(x)}, the “kernel” of the dlog(G,s) -map. It
]
is a finite-dimensional differential algebraic group. K(G,s)
denotes the differential
field generated over K by G∂ (K dif f ), the points of G∂ in the differential closure
K dif f of K. As any two solutions y1 , y2 ∈ G(K dif f ) of a logarithmic differential
equation dlog(G,s) (−) = A on (G, s) over K, differ by an element of G∂ (K dif f ), it
]
]
follows that tr.deg(K(G,s)
(y)/K(G,s)
) is the same, for any solution of the equation
dif f
in G(K
).
We write the lemma below for the case where G is commutative and use additive
notation, although suitably written it holds in general.
Lemma 3.3. Let (G, s) be a commutative connected algebraic δ-group over K where
K is algebraically closed. Let A ∈ LG(K) and let y ∈ G(K dif f ) be a solution of
the equation
(*) dlog(G,s) (−) = A.
Then the following are equivalent:
]
]
) equals min{dim(H) : H is a connected algebraic
(y)/K(G,s)
(i) tr.deg(K(G,s)
δ-subgroup of G defined over K and A ∈ L(H) + dlog(G,s) (G(K))}.
(ii) For some solution y0 of (*), K(y0 ) is a G∂ -strongly normal extension of
K.
Proof. The condition in (i) that A ∈ L(H)+dlog(G,s) (G(K)) is obviously equivalent
to y ∈ H + G(K) + G∂ (K dif f ) and will be used that way.
(i) ⇒ (ii): Fix a solution y of (*). Suppose that H is a connected algebraic δsubgroup of G such that y ∈ H + G(K) + G∂ (K dif f ) and
]
]
dim(H) = tr.deg.(K(G,s)
(y)/K(G,s)
).
So there is y0 ∈ H +G(K) of the form y +g for some g ∈ G∂ (K dif f ). Note that y0 is
also a solution of (*) and moreover tr.deg(K(y0 )/K) ≤ dim(H). On the other hand,
]
]
]
]
tr.deg.(K(G,s)
(y0 )/K(G,s)
) = tr.deg.(K(G,s)
(y)/K(G,s)
) = dim(H), by assumption.
]
The conclusion is that tr.deg(K(y0 )/K) = dim(H) and y0 is independent from KG
]
∂
dif f
over K (in DCF0 ). As KG = K(G (K
)), the latter says that y0 is independent
from G∂ (K dif f ) over K, and this says that y0 is independent from G∂ over K. As
K is algebraically closed and G∂ is definable, this says that tp(y0 /K) has a unique
extension to a complete type over K ∪ G∂ which is precisely condition (II). As y0
is a solution of (*) we have condition (I) too.
Conversely, suppose (ii) holds, witnessed by y. By (II), y is independent of G∂
over K. By [1], Remark following Fact 2.2, let H be a connected algebraic δsubgroup of G, over K such that H ∂ = Aut(Y /K, G∂ ). Consider the orbit y + H ∂ .
It is defined over K ] , and over K(y) (in DCF0 ), so also over K. But then y +H, the
Zariski closure of y+H ∂ is defined over K (in DCF0 ). By elimination of imaginaries
in ACF, let e be a code of y + H (in ACF). But then e ∈ K. So y + H is defined
over K in ACF so has a K-rational point y1 ∈ G(K). So y ∈ H + G(K). The
minimality of dim(H) is clear.
Conjecture 2.3 from [1] can now be restated as: Let G be an almost semiabelian δgroup over K = C(t)alg , let A ∈ LG(K), and consider the equation dlog(G,s) (−) =
A. Then the equation has a solution y such that K(y) is a G∂ -strongly normal
extension of K.
12
OMAR LEÓN SÁNCHEZ AND ANAND PILLAY
Finally let us mention an example, even a Picard-Vessiot extension, where the
Galois correspondence is not direct.
Example 3.4. We work in the ordinary case. Let K = Q and L = Q(et ), with
derivation d/dt. Then L is a Picard-Vessiot extension of K for the equation δy = y.
It can be seen that in this case G = Gm (C). The intermediate differential fields are
Q(ent ) as n varies in N. However, the only algebraic subgroups of G(K) = Gm (Q)
are the two obvious ones and {−1, 1}.
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Omar León Sánchez, McMaster University, Department of Mathematics and Statistics, 1280 Main St W, Hamilton, ON, L8S 4L8.
E-mail address: [email protected]
Anand Pillay, University of Notre Dame, Department of Mathematics, 255 Hurley,
Notre Dame, IN, 46556.
E-mail address: [email protected]